SOME OF THE RESULTS OBTAINED IN THE YAROSLAVL BRANCH ALGEBRAIC SCHOOL OF M. D. GRINDLINGER
https://doi.org/10.22405/2226-8383-2014-15-4-5-31
Abstract
We review the main results obtained in the Yaroslavl branch of Martin Greendlinger’s algebraic school from the middle 1970s up to the present.
About the Authors
V. G. Durnev
Ярославский государственный университет им. П. Г. Демидова
Russian Federation
Russian Federation
O. V. Zetkina
Ярославский государственный университет им. П. Г. Демидова
Russian Federation
Russian Federation
References
1. Quine, W. 1946, "Concatenation as a basis for arithmetic" , J. Symbolic Logic, vol. 11. pp. 105 – 114.
2. Durnev, V. G. 1970, "On positive formulas on groups" , Proc. of Math. Dept. of Tula State Pedagogical Inst., Ser. Geometry and Algebra. [Uchtnye Zapiski Matematicheskoy Kafedry Tul’skogo Pedogogicheskogo Instituta], no. 2. pp. 215 – 241. (Russian)
3. Durnev, V. G. 1972, "On positive theory of free semigroup" , Groups and semigropus, Tula, pp. 122 – 172 (Russian). 4. Durnev, V. G. 1973, "Positive theory of free semigroup" , Doklady Akademii Nauk SSSR, vol. 211, no. 4. pp. 772 – 774. (Russian)
4. Durnev, V. G. 1974, "Positive formulas in free semigroups" , Siberian Mathematical Journal, vol. 15, issue 5, pp. 796–800 (Translated from Sibirskii Matematicheskii Zhurnal, vol. 15, no. 5, pp. 1131–1137) [doi: 10.1007/BF00966439]
5. Durnev, V. G. 1973, "Positive theories of free semigroups" , Ph D. Thesis, Moscow State Pedagogical Institution. (Russian)
6. Marchenkov, S. S. 1982, "Undecidability of the positive ∀∃-theory of a free semigroup" , Siberian Mathematical Journal, vol. 23, no. 1. pp. 196 – 198. (Russian)
7. Durnev, V. G. 1995, "Undecidability of the positive ∀∃3 -theory of a free semigroup" , Siberian Mathematical Journal, vol. 36, no. 5. pp. 917–929 (Translated from: Sibirskii Matematicheskii Zhurnal, vol. 36, no. 5, pp. 1067 – 1080) [doi: 10.1007/BF02112533].
8. Durnev, V. G. 2000, "Undecidability of a simple fragment of a positive theory with a single constant for a free semigroup of rank two" , Mathematical Notes, vol. 67, no. 2, pp. 191–200. (Russian)
9. Hmelevski˘i, Ju. I. 1971, "Equations in a free semigroup" , Proceedings of the Steklov Institute of Mathematics, vol. 107, pp. 1–270. (Russian)
10. Matiyasevich, Yu. V. 1968, "The connection between systems of equations in words and lengths with Hilbert’s 10th problem" , Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI), vol. 8. pp. 132 – 143. (Russian)
11. Kosovski˘i, N. K. 1972, "Certain properties of the solutions of equations in a free semigroup. Investigations in constructive mathematics and mathematical logic" , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 32, pp. 21–28, 154. (Russian)
12. Kosovski˘i, N. K. 1972, "On sets represented as solutions of equations in words and lengths" , 2nd USSR Conference on mathematical logics. Moscow. Book of abstracts, pp. 23. (Russian)
13. Kosovski˘i, N. K. 1974, "The solution of systems that consist simultaneously of word equations and word length inequalities" , Investigations in constructive mathematics and mathematical logic, VI (dedicated to A. A. Markov on the occasion of his 70th birthday), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 40, pp. 24–29, 156. (Russian)
14. Durnev, V. G. 1974, "On equations in free semigroups and groups" , Mathematical Notes, vol. 16, no. 5. pp. 717 – 724. (Russian)
15. B¨uchi, J. R. & Senger, S. 1988, "Definability in the existential theory of concatenation and undecidable extensions of this theory" , Z. Mat. Log. und Grundl. Math., vol. 34, no. 4. pp. 337 – 342.
16. B¨uchi, J. R. & Senger, S. 1986/87, "Coding in the existential theory of concatenation" , Arch. Math. Logik Grundlag., vol. 26, pp. 101 – 106.
17. Senger, S. 1982, "The Existential Theory of concatenation" , Ph. D. Dissertation, Purdue University.
18. Durnev, V. G. 1977, "On some equations on free semigropus" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. pp. 57 – 59. (Russian)
19. Makanin, G. S. 1977, "The problem of the solvability of equations in a free semigroup" , Dokl. Akad. Nauk SSSR, vol. 233, no. 2, pp. 287 – 290. (Russian)
20. Makanin, G. S. 1977, "The problem of solvability of equations in a free semigroup" , Math. USSR Sb., vol. 32. no. 2, pp. 129–198. (Translated from: Math. Sbornik. 1977. Vol. 103(145), no. 2(6). P. 147–236.) [doi:10.1070/SM1977 v032n02ABEH002376]
21. Durnev, V. G. & Zetkina, O. V. 2003, "On equations in free semigroups with certain constraints on their solutions" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
22. Durnev, V. G. & Zetkina, O. V. 2008, "Equations with language constraints to their solutions in free monoids" , Mathematics, cybernetics, informatics — Prof. A. Yu. Levin Memorial Conference. Yaroslavl’ State University., pp. 93 – 99. (Russian)
23. Durnev, V. G. & Zetkina, O. V. 2008, "On equations in free semigroups with certain constraints on their solutions" , Zapiski Nauchnykh Seminarov POMI (Proc. of St.-Petersburg branch of V. A. Steklov Math. Inst.), St.-Petersburg., vol. 358, pp. 120 – 129. (Russian)
24. Durnev, V. G. & Zetkina, O. V. 2008, "On equations in free semigroups with certain constraints to their solutions" , Journal of Mathematical Sciences., vol. 158, no. 5, pp. 671 – 676. (Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 120-–129) [doi:10.1007/s10958-009-9409-z]
25. Durnev, V. G. & Zetkina, O. V. 2010, "Equations with subsemigroup constraints to their solutions in free semigroups " , Chebyshevskii Sb., vol. XI, issue 3(35), pp. 78 – 87. (Russian)
26. Durnev, V. G. 1991, "On equations with endomorphisms in free semigroups and groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’., pp. 30 – 35. (Russian)
27. Durnev, V. G. 1992, "Equations with endomorphisms in free semigroups" , Diskret. Mat.. vol. 4, no. 2, pp. 136 – 141. (Russian)
28. Durnev, V. G. 1992, "On equations in words and lengths with endomorphisms" , Russian Mathematics (Izvestiya VUZ. Matematika). vol. 36, no. 8, pp. 26 – 30. (Russian)
29. Whitehead, J. H. C. 1936, "On equivalent sets of elements in a free group" Proc. London Math. Soc., vol. 37, pp. 782 – 800.
30. Durnev, V. G. 2003, "NP-completeness of the problem of endomorphic reducibility for elements of a free semigroup of an countable rank" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
31. Schupp, P. E. 1969, "On the substitution problem for free groups" , Proc. Amer. Math. Soc., vol. 23, no. 2, pp. 421 – 423.
32. Edmunds, C. C. 1975, "On the endomorphisms problem for free group" , Com. Algebra., no. 3, pp. 7 – 20.
33. Lyndon, R. C. 1960, "Equations in free groups" , Trans. Amer. Math. Soc., vol. 96, pp. 445 – 457.
34. Lyndon, R. C. 1966, "Dependence in groups" , Colloq. Math., no. 4, pp. 275 – 283.
35. Hmelevski˘i, Ju. I. 1971, "Systems of equations in a free group I" , Math. USSRIzv., vol. 5, no. 6, pp. 1245 –1276. (Translated from Izv. AN USSR. Ser. Mathem. 1971. Vol. 35, no. 6. P. 1237–1268.) [doi:10.1070/IM1971v005n06ABEH001234]
36. Makanin, G. S. 1983, "Equations in a free group" , Math. USSR-Izv., vol. 21, no. 3, pp. 483–546. (Translated from Izv. AN USSR. Ser. Mathem. 1982. Vol. 46, no. 6. P. 1199 – 1274.) [doi:10.1070/IM1983v021n03ABEH001803]
37. Makanin, G. S. 1985, "Decidability of the universal and positive theories of a free group" , Math. USSR-Izv., vol. 25, no. 1, pp. 75–88. (Translated from Izv. AN USSR. Ser. Mathem. 1984. Vol. 48, no. 4. P. 735 – 749.) [doi:10.1070/ IM1985v025n01ABEH001269]
38. Razborov, A. A. 1984, "Systems of equations in a free group" , Izv. Akad. Nauk SSSR Ser. Mat., vol. 48, no. 4, pp. 779 – 832. (Russian) [URL:http: //www.ams.org/mathscinet-getitem?mr=755958]
39. Malkhasyan, A. Sh. 1986, "On solvability in subgroups of equations in a free group" , Applied mathematics. Issue 2, pp. 42 – 47. (Russian)
40. Durnev, V. G. 1990, "On one generalization of the Problem 9.25 from the “Kourovka notebook”" , Mathematical Notes., vol. 47, no. 2, pp. 117–121. (Translated from Matematicheskie Zametki. 1990. Vol. 47, no. 2. P. 15 – 19.) [doi:10.1007/BF01156819]
41. Durnev, V. G. 1993, "Equations with constraints on the solution in free groups" , Mathematical Notes., vol. 53, no. 1, pp. 26—29. (Translated from Matematicheskie Zametki. Vol. 53, no. 1. P. 36 – 40.) [doi:10.1007/BF01208519]
42. Durnev, V. G. 1995, "On equations with subgroup constraints on solutions in free groups " , Discrete Math. Appl., vol. 5, no. 6, pp. 567—575. (Translated from: Diskret. Mat. 1995. Vol. 7. no. 4, 60–67.) [doi:10.1515/dma.1995.5.6.567]
43. Durnev, V. G. & Zetkina, O. V. 2006, "On equations with subgroup constraints on solutions in free groups" , Mathematics in Yaroslavl’ University (30th aniversary of Math. Faculty). Yaroslavl’, pp. 181 – 200. (Russian)
44. Durnev, V. G. & Zetkina, O. V. 2010, "On some equations with with constraints to their solutions" , Chebyshevskii Sb., vol. XI, issue 3(35), pp. 88 – 97. (Russian)
45. Durnev, V. G. 1996, "On the solvability problem for equations with a single coefficient" , Mathematical Notes., vol. 59, no. 6, pp. 601–610. (Translated from Matematicheskie Zametki, 1996. Vol. 59. no. 6. P. 832 – 845). [doi:10.1007/ BF02307209]
46. Durnev, V. G. & Zetkina, O. V. 2012, "On the equations resolved with respect to variables in free groups with constraints to the solutions" , Chebyshevskii Sb., vol. XIII, issue 1(41), pp. 63 – 80. (Russian)
47. Durnev V. G. & Zetkina, O. V. 2012, "NP-complexity of the decidability problem for the equations with right-hand-side in a free group" , Chebyshevskii Sb., vol. XIII, issue 1(41), pp. 46 – 53. (Russian) 49. Coulbois, T. & Khelif, A. 1999, "Equations in free groups are not finitely approximable" , Proceedihgs of the American mathematical society., vol. 127, no. 4, pp. 2435 – 2436.
48. Durnev, V. G. 2012, "On equations in free groups" , Chebyshevskii Sb., vol. XIII, issue 1(41), pp. 59 – 62. (Russian)
49. Durnev, V. G. 2012, "On one A. I. Mal’cev’s question from the “Kourovka notebook”" , Chebyshevskii Sb., vol. XIII, issue 1(41), pp. 54 – 58. (Russian)
50. Mazurov, V. D. & Khukhro, E. I. (Eds) 2014, "Kourovka notebook (non-solved problems of the group theory). Ed.18th" , Novosibirsk. (Russian)
51. Durnev, V. G. 1981, "On systems of equations on free nilpotent groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’, pp. 66 – 69. (Russian)
52. Mal’cev, A. I. 1960, "Some correspondences between rings and groups", Math. Sbornik (N.S.), vol. 50(92), no. 2, pp. 257 – 266. (Russian) [URL:http:// mi.mathnet.ru/msb4792]
53. Roman’kov, V. A. 1977, "Unsolvability of the endomorphic reducibility problem in free nilpotent groups and in free rings" Algebra and Logic., vol. 16, no. 4, pp. 310 – 320. (Translated from Algebra i Logika. 1977. Vol. 16, no. 4. P. 457 – 471). [doi:10.1007/BF01669283]
54. Roman’kov, V. A. 1979, "Universal theory of nilpotent groups" , Mathematical notes of the Academy of Sciences of the USSR., vol. 25, no. 4, pp. 253 – 258. (Translated from Matematicheskie Zametki. 1979. Vol. 25, no. 4. P. 487 – 496). [doi:10.1007/BF01688474]
55. Repin, N. N. 1988, "Some simply defined groups without a decidability-testing algorithm" , Cybernetics, Calculation Complexity and Applied Mathematical Logics. Moscow., pp. 167 – 174. (Russian)
56. Spielrein, V. E. 1990, "On equations in the groups of F/γn(R) type" Algorithmic problems of groups and semigroups. Tula., pp. 164 – 183. (Russian)
57. Durnev, V. G. 1988, "Undecidability of endomorphic reducibility problem for sets of elements of a free nilpotent group" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’., pp. 56 – 63. (Russian)
58. Durnev, V. G. & Zetkina, O. V. 2003, "On fragments of elementary theories of free nilpotent groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
59. Durnev, V. G. 1989, "The Mal’tsev-Nielsen equation in a free metabelian group of rank two" , Mathematical notes of the Academy of Sciences of the USSR., vol. 46, Issue 6, pp. 927–929. (Translated from Matematicheskie Zametki, vol. 46, No. 6, pp. 57–60, December, 1989) [doi:10.1007/BF01158628]
60. Maltsev, A. I. 1962, "On the equation zxyx−1 y −1 z −1 = aba−1 b −1 in a free group" , Algebra i Logika., vol. 1, no. 5, pp. 45 – 50. (Russian)
61. Maltsev, A. I. 1960, "On free decidable groups" , USSR Doklady Ser. Math., vol. 130, no. 3, pp. 495 – 498. (Russian)
62. Roman’kov, V. A. 1979, "Equations in free metabelian groups" , Siberian Mathematical Journal., vol. 20, no. 3, pp. 469–471. (Translated from Sibirskii Matematicheskii Zhurnal, vol. 20, No. 3, pp. 671—673, 1979) [doi:10.1007/ BF00969959]
63. Durnev, V. G. 1992, "Undecidability of a positive ∃-theory with one constant for a free solvable group" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’., pp. 30 – 35. (Russian)
64. Merzlyakov, Yu. I. 1966, "Positive formulae on free groups" , Algebra i Logika., vol. 5, no. 4, pp. 25 – 42. (Russian)
65. Durnev, V. G. 1990, "On Tarski problem for free groups of some manifolds" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’., pp. 25 – 35. (Russian)
66. Lawrence, J. 1986, "Tarski’s problem for varieties of groups with commutator identity" , J. Symbolic Logic., vol. 51, no. 1, pp. 75 – 78.
67. Rogers, P., Smith, H. & Solitar, D. 1986, "Tarski’s problem for solvable groups" , Proc. Amer. Math. Soc., vol. 96, no 4, pp. 668 – 671.
68. Sacerdote, G. S. 1973, "Almost all free products of groups have the same positive theory" , J. Algebra., vol. 27, no. 3, pp. 475 – 485.
69. Sacerdote, G. S. 1973, "Elementary properties of free groups" , Trans. Amer. Math. Soc., vol. 178, pp. 127 – 138.
70. Maltsev, A. I. 1961, "On elementary properties of linear groups" , Problems of Mathematics and Mechanics. Novosibirsk., pp. 110 – 132. (Russian)
71. Durnev, V. G. & Kazarin, L. S. 1994 , "On unversal theories of some groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
72. Durnev, V. G. 1994, "Undecidability of some bounded theories of групп SL(n, Z) and GL(n, Z) (n > 3) groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
73. Durnev, V. G. 1995, "On elementary theories of integer linear groups" , Izvestiya: Mathematics, vol. 59, no. 5, pp. 919–934. [doi:10.1070/IM1995v059n 05ABEH000040]
74. Lyndon, R. C. & Schupp, P. E. 2001, "Combinatorial Group Theory" , Springer. 339 p.
75. Durnev, V. G. 1998, "Semigroups with identities of some F-groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
76. Durnev, V. G., Zetkina, O. V. & Zetkina, A. I. 2014, "On the Tits’ alternative for subgroups of F-groups" , Chebyshevskii Sb., vol. 15, issue 1(49), pp. 110—120. (Russian). URL:http://mi.mathnet.ru/eng/cheb15]
77. Durnev, V. G. 1987, "On width of the commutant of B3 и B4 braid groups" , XIX USSR algebraic conference. Book of abstracts. L’vov. (Russian)
78. Repin, N. N. 1986, "On commmutator equations in B3 and B4 groups" , Algorithmic problems of group and semigroup theory. Tula., pp. 114 – 117. (Russian)
79. Roman’kov, V. A. 1982, "Width of verbal subgroups in solvable groups" , Algebra and Logic., vol. 21, no. 1, pp. 41–49. (Translated from Algebra i Logika, Vol. 21, no. 1, pp. 60–72, 1982). [doi:10.1007/BF01987820]
80. Rhemtulla, A. H. 1961, "Commutators of certain finitely generated Solvable groups" , Can. J. Math., vol. 21, no. 5, pp. 1160 – 1164.
81. Rhemtulla, A. H. 1968, "A problem of bounded expressibility in free products" , Proc. of the Cambridge Phil. Soc., vol. 64, no. 3, pp. 573 – 584.
82. Durnev, V. G. & Zetkina, O. V. 1994, "On factor-groups of B(n) braids and linear SL(n, Z) and GL(n, Z) groups" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
83. Durnev, V. G. 1988, "On formulae with a length function on a free group" , 10th USSR Conference on math. logics. Book of abstracts. Leningrad., pp. 56 (Russian)
84. Durnev, V. G. & Zetkina, O. V. 1991, "On a positive theory of a free group in a signature extended by a length function" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’., pp. 25 – 29. (Russian)
85. Huber-Dyson, V. 1974, "The Undecidability of Free Groups with a Length Function" , University of Calgary, Mathematics Research Paper., no. 221, pp. 1 – 26.
86. Myasnikov, A. G. & Remeslennikov, V. N. 1987, "Elementary equivalence of free products" , Preprint of USSR Acad. Sci. Siberian branch., Comput. Centre, Novosibirsk., no. 718. 20 p. (Russian)
87. Durnev, V. G. & Zetkina, O. V. 1994, "Algoritmically undecidable problems for Diophantine sets in Π2" , Group Theory & Homological Algebra, Yaroslavl’ State University. Yaroslavl’. (Russian)
88. Durnev, V. 1997, "Studying Algorithmic Problems for Free Semigroups and Groups" , Lecture Notes in Computer Science., vol. 1234, pp. 88 – 101. (Russian)
89. Durnev, V. G. 1997, "Studying Algorithmic Problems for Free Semigroups and Groups": Dr. Sci. Thesis, Moscow State University. (Russian)
90. Bardakov, V. G. 1993, "On the theory of braid groups" , Russian Academy of Sciences. Sbornik Mathematics., vol. 76, no. 1, pp. 123–153. [doi: 10.1070/SM1993 v076n01ABEH003404]
91. Adian, S. I. & Durnev, V. G. 2000, "Decision problems for groups and semigroups" , Russian Mathematical Surveys., vol. 55, no. 2, pp. 207. [doi: 10.1070/ RM2000v055n02ABEH000267]
Review
For citations:
Durnev V.G., Zetkina O.V. SOME OF THE RESULTS OBTAINED IN THE YAROSLAVL BRANCH ALGEBRAIC SCHOOL OF M. D. GRINDLINGER. Chebyshevskii Sbornik. 2014;15(4):5-31. (In Russ.) https://doi.org/10.22405/2226-8383-2014-15-4-5-31
Views:
485
Email this article 